Numerical experiments using the barycentric Lagrange treecode to compute correlated random displacements for Brownian dynamics simulations

Abstract

To account for hydrodynamic interactions among solvated molecules, Brownian dynamics simulations require correlated random displacements $g=D^{1/2}z$, where D is the $3N\times 3N$ Rotne-Prager-Yamakawa diffusion tensor for a system of N particles and z is a standard normal random vector. The Spectral Lanczos Decomposition Method (SLDM) computes a sequence of Krylov subspace approximations $g_k\to g$, but each step requires a dense matrix-vector product $Dq$ with a Lanczos vector q, and the $O\left(N^2\right)$ cost of computing the product by direct summation (DS) is an obstacle for large-scale simulations. This work employs the barycentric Lagrange treecode (BLTC) to reduce the cost of the matrix-vector product to $O(N\log N)$ while introducing a controllable approximation error. Numerical experiments compare the performance of SLDM-DS and SLDM-BLTC in serial and parallel (32 core, GPU) calculations.